L(s) = 1 | + 1.41·2-s + 0.356i·3-s + 2.00·4-s + 2.23i·5-s + 0.504i·6-s + 2.82·8-s + 2.87·9-s + 3.16i·10-s + 0.712i·12-s − 0.796·15-s + 4.00·16-s + 4.06·18-s + 4.47i·20-s − 7.50·23-s + 1.00i·24-s − 5.00·25-s + ⋯ |
L(s) = 1 | + 1.00·2-s + 0.205i·3-s + 1.00·4-s + 0.999i·5-s + 0.205i·6-s + 1.00·8-s + 0.957·9-s + 1.00i·10-s + 0.205i·12-s − 0.205·15-s + 1.00·16-s + 0.957·18-s + 1.00i·20-s − 1.56·23-s + 0.205i·24-s − 1.00·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.96002 + 1.35228i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.96002 + 1.35228i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 0.356iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 7.50T + 23T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 12.6iT - 41T^{2} \) |
| 43 | \( 1 - 9.10T + 43T^{2} \) |
| 47 | \( 1 + 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.6iT - 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 18.2iT - 83T^{2} \) |
| 89 | \( 1 - 14.1iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.25390539638502825866042635935, −9.683677315665473801490198707145, −8.119519577385236603924988126532, −7.38047161651305375457980976267, −6.56399249885958802688380055631, −5.90275518654556975089505135498, −4.66529437501558679553817407982, −3.91321586130013769922360995916, −2.93633605386391900586044834974, −1.80566340343590033790814014928,
1.25268962078996686745917344808, 2.38086572058216006811791570132, 3.94737995531939009432174021289, 4.44447595784280605716840807434, 5.50324899823337689741015344716, 6.25910118586012039297725883072, 7.33368889414521224192360224106, 7.967856974785242192645326261898, 9.052082578461296865822408042657, 10.05274986073419237339197690931